Signals and Spectral Methods in Geoinformatics Lecture 2: Fourier Series

Development of a function defined in an interval into Fourier Series Jean Baptiste Joseph Fourier

REPRESENTING A FUNCTION BY NUMBERS f (t) t Τ coefficients α1, α2, ... of the function f = a1φ1+ a2 φ2 + ... known base functions φ1, φ2, ... function f

The base functions of Fourier series +1 –1 Τ +1 +1 +1 +1 –1 –1 –1 –1 Τ Τ Τ Τ +1 +1 +1 +1 –1 –1 –1 –1 Τ Τ Τ Τ

Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms:

Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: Every base function has: period:

Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: Every base function has: period: frequency:

Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: Every base function has: period: frequency: angular frequency:

Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: Every base function has: period: frequency: angular frequency: fundamental period fundamental frequency fundamental angular frequency

term angular frequencies Development of a real function f(t) defined in the interval [0,T ] into Fourier series 3 alternative forms: Every base function has: period: frequency: angular frequency: fundamental period fundamental frequency fundamental angular frequency term periods term frequencies term angular frequencies

Development of a real function f(t) defined in the interval [0,T ] into Fourier series simplest form:

Development of a real function f(t) defined in the interval [0,T ] into Fourier series simplest form: Fourier basis (base functions):

Development of a real function f(t) defined in the interval [0,T ] into Fourier series simplest form: Fourier basis (base functions):

An example for the development of a function +1 –1 f (x) An example for the development of a function in Fourier series Separate analysis of each term for k = 0, 1, 2, 3, 4, …

+1 –1 f (x) k = 0 base function +1 –1

+1 –1 f (x) k = 0 contribution of term +1 –1

+1 –1 f (x) k = 1 base functions +1 –1 +1 –1

+1 –1 f (x) k = 1 contributions of term +1 –1 +1 –1

+1 –1 f (x) k = 2 base functions +1 –1 +1 –1

+1 –1 f (x) k = 2 contributions of term +1 –1 +1 –1

+1 –1 f (x) k = 3 base functions +1 –1 +1 –1

+1 –1 f (x) k = 3 contributions of term +1 –1 +1 –1

+1 –1 f (x) k = 4 base functions +1 –1 +1 –1

+1 –1 f (x) k = 4 contributions of term +1 –1 +1 –1

+1 –1 f (t)

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product:

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

Orthogonality of Fourier base functions Inner product of two functions: Fourier basis: Orthogonality relations (km): Norm (length) of a function:

Computation of Fourier series coefficients Ortjhogonality relations (km):

Computation of Fourier series coefficients Ortjhogonality relations (km):

Computation of Fourier series coefficients Ortjhogonality relations (km): 0 for km 0 for km

Computation of Fourier series coefficients Ortjhogonality relations (km): 0 for km 0 for km

Computation of Fourier series coefficients Ortjhogonality relations (km): 0 for km 0 for km

Computation of Fourier series coefficients Ortjhogonality relations (km): 0 for km 0 for km

Computation of Fourier series coefficients of a known function:

Computation of Fourier series coefficients

Computation of Fourier series coefficients change of notation

Computation of Fourier series coefficients change of notation

Computation of Fourier series coefficients

Computation of Fourier series coefficients

Alternative forms of Fourier series (polar forms) Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk ! ρk = «length» θk = «azimuth» φk = «direction angle» φk + θk = 90

Alternative forms of Fourier series (polar forms) Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk ! ρk = «length» θk = «azimuth» φk = «direction angle» φk + θk = 90

Alternative forms of Fourier series (polar forms) Polar coordinates ρk, θk or ρk, φk, from the Cartesian ak, bk ! ρk = «length» θk = «azimuth» φk = «direction angle» φk + θk = 90

Alternative forms of Fourier series (polar forms)

Alternative forms of Fourier series (polar forms)

Alternative forms of Fourier series (polar forms) θk = phase (sin)

Alternative forms of Fourier series (polar forms) θk = phase (sin)

Alternative forms of Fourier series (polar forms) θk = phase (sin) φk = phase (cosine)

Fourier series for a complex function Fourier series of real functions: «real» part «imaginary» part

Fourier series for a complex function Fourier series of real functions: «real» part «imaginary» part setting

Fourier series for a complex function Fourier series of real functions: «real» part «imaginary» part setting

Fourier series for a complex function Implementation of complex symbolism:

Fourier series for a complex function Implementation of complex symbolism:

Fourier series for a complex function Implementation of complex symbolism:

Fourier series for a complex function Implementation of complex symbolism:

Fourier series for a complex function Implementation of complex symbolism:

Complex form of Fourier series Development of a complex function into a Fourier series with complex base functions and complex coefficients Computation of complex coefficients for a known complex function

Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Ortjhogonality of the complex basis Conjugateς z* of a complex number z : inner product:

Fourier series of a real function using complex notation Implementation of complex symbolism:

Fourier series of a real function using complex notation Implementation of complex symbolism:

Fourier series of a real function using complex notation Implementation of complex symbolism:

Fourier series of a real function using complex notation Implementation of complex symbolism:

Fourier series of a real function using complex notation Implementation of complex symbolism:

Fourier series of a real function using complex notation

Fourier series of a real function using complex notation f (t) = real function

Fourier series of a real function using complex notation f (t) = real function

Fourier series of a real function using complex notation f (t) = real function

Fourier series of a real function using complex notation f (t) = real function

Fourier series of a real function using complex notation f (t) = real function The imaginary part disappears !

Fourier series of a real function: Real and complex form

Fourier series of a real function: Real and complex form

Fourier series of a real function: Real and complex form

Fourier series of a real function: Real and complex form

Fourier series of a real function: Real and complex form

Extension of the function f (t) outside the interval [0, T ] 2T 3T –T –2T The extension is a periodic function, with period Τ for every integer n CAUSE OF USUAL MISCONCEPTION: “Fourier series expansion deals with periodic functions»

Fourier series on the circle (naturally periodic domain) θ (angle)

Fourier series on the circle (naturally periodic domain) θ (angle)

Fourier series on the circle (naturally periodic domain) θ (angle)

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Base functions: Ty Tx

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Base functions: (angular frequencies along x and y ) Ty Tx

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Base functions: (angular frequencies along x and y ) Ty Tx

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Base functions: (angular frequencies along x and y ) Ty Tx

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty)

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Equivalent to double Fourier series: First along x and then along y (or vice-versa)

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Equivalent to double Fourier series: First along x and then along y (or vice-versa)

Fourier series on the plane Expansion of function f (x,y) inside an orthogonal parallelogram (0  x  Tx, 0  y  Ty) Equivalent to double Fourier series: First along x and then along y (or vice-versa)

Fourier series on the plane Inner product: for every Α = a,b,c,d and B = a,b,c,d Orthogonal Fourier basis ! ή

Fourier series on the plane Inner product: for every Α = a,b,c,d and B = a,b,c,d Orthogonal Fourier basis ! ή Computation of coefficients:

Fourier series on the plane Complex form:

Fourier series on the plane Complex form: Fourier series in n dimensions

Fourier series in n dimensions

Fourier series in n dimensions In matrix notation: domain of definition: (orthogonal hyper-parallelepiped) (parallelepiped volume)

Fourier series in n dimensions In matrix notation: domain of definition: (orthogonal hyper-parallelepiped) (parallelepiped volume)

Fourier series on any interval [Α, Β]

Approximating a function by a finite Fourier series expansion Question : What is the meaning of the symbol in the Fourier series expansion?

Approximating a function by a finite Fourier series expansion Question : What is the meaning of the symbol in the Fourier series expansion? Certainly not that infinite terms must be summed! This is impossible!

Approximating a function by a finite Fourier series expansion Question : What is the meaning of the symbol in the Fourier series expansion? Certainly not that infinite terms must be summed! This is impossible! In practice we can use only a finite sum with a «sufficiently large» integer Ν

Approximating a function by a finite Fourier series expansion Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – fN(t)|| < ε

Approximating a function by a finite Fourier series expansion Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – fN(t)|| < ε Attention: || f(t) – fN(t)|| < ε does not necessarily mean that the difference | f(t) – fN(t)| is small for every t !!!

Approximating a function by a finite Fourier series expansion Sufficiently large Ν means: For whatever small ε > 0 there exists an integer Ν such that || f(t) – fN(t)|| < ε Attention: || f(t) – fN(t)|| < ε does not necessarily mean that the difference | f(t) – fN(t)| is small for every t !!! It would be desirable (though not plausible) that max | f(t) – fN(t)| < ε in the interval [0,Τ]

Characteristics of the Fourier series expansion  The coefficients ak, bk become generally smaller as k increases

Characteristics of the Fourier series expansion  The coefficients ak, bk become generally smaller as k increases  The base functions cosωkt, sinωkt have larger frequency ωk = kωT and smaller period Τk = T/k (i.e. more detail) as k increases

Characteristics of the Fourier series expansion  The coefficients ak, bk become generally smaller as k increases  The base functions cosωkt, sinωkt have larger frequency ωk = kωT and smaller period Τk = T/k (i.e. more detail) as k increases  The terms [ak cosωkt + bk sinωkt] have a more detailed contribution to fN(t) a k increases

Characteristics of the Fourier series expansion  The coefficients ak, bk become generally smaller as k increases  The base functions cosωkt, sinωkt have larger frequency ωk = kωT and smaller period Τk = T/k (i.e. more detail) as k increases  The terms [ak cosωkt + bk sinωkt] have a more detailed contribution to fN(t) a k increases  As Ν increases more details are added to the Fourier series expansion

Characteristics of the Fourier series expansion  The coefficients ak, bk become generally smaller as k increases  The base functions cosωkt, sinωkt have larger frequency ωk = kωT and smaller period Τk = T/k (i.e. more detail) as k increases  The terms [ak cosωkt + bk sinωkt] have a more detailed contribution to fN(t) a k increases  As Ν increases more details are added to the Fourier series expansion  For a sufficient large Ν (which?) fN(t) ia a satisfactory approximation to f(t) within a particular application

as the best approximation of a function within an interval The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question : In an expansion with finite number of terms Ν, of the form which are the values of the coefficients Α0, Ak, Bk for which the sum fN(t) best approximates f(t), in the sense that

as the best approximation of a function within an interval The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question : In an expansion with finite number of terms Ν, of the form which are the values of the coefficients Α0, Ak, Bk for which the sum fN(t) best approximates f(t), in the sense that Answer : The Fourier coefficients a0, ak, bk

as the best approximation of a function within an interval The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question : What is the meaning of the symbol in the Fourier series expansion?

as the best approximation of a function within an interval The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question : What is the meaning of the symbol in the Fourier series expansion? ANSWER : The symbol means that we can choose a sufficiently large Ν, so that we can make satisfactorily small the error δf(t) = f(t) – fN(t)

as the best approximation of a function within an interval The finite sum of the Fourier series expansion as the best approximation of a function within an interval Question : What is the meaning of the symbol in the Fourier series expansion? ANSWER : The symbol means that we can choose a sufficiently large Ν, so that we can make satisfactorily small the error δf(t) = f(t) – fN(t) Specifically: For every small ε there exists a corresponding integer Ν = Ν(ε) such that small mean square error !

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